Semi-Free Circle Actions on Spin${}^c$-Manifolds

Abstract

When a compact Lie group acts differentiably on smooth manifolds, various results have been known concerning the characteristic numbers of the manifolds. The most frequently used tool is AtiyahSinger Lefschetz formula [2]. However, several approaches have been made to obtain similar results by geometric methods. Hattori and Taniguchi [6] investigated the cobordism groups of oriented or weakly almost complex manifolds with S-actions and recovered Kosniowski formula [8] and Atiyah-Singer formula [2]. But as for Spinmanifolds, no cobordism theoretic interpretation of Atiyah-Hirzebruch theorem [3] has been known so far. In this paper we consider Spin-manifolds with semi-free S-actions. By purely geometric methods, we obtain Todd genus formula which relates the Todd genus of the manifold and the local behaviour of the S-action around the fixed point sets. A similar formula has been given by Petrie [9] using Atiyah-Singer Lefschetz formula and the Dirac operator. As applications of our Todd genus formula, we can prove the results of Kosniowski [8] and Atiyah-Hirzebruch [3] in the semi-free case.